In a science class lab, a mass is suspended from a spring above a tabletop. The mass is pulled down and released. By examining frames from a video as the mass bobs up and down, the class obtains the model h(t)=5 $\sin 4(t-0.39)+10$ for the height h of the mass above the tabletop at time $t$ in seconds. Cassandra notices that after several seconds the mass is not bobbing up and down as far, but it still seems to take about the same time for a cycle. Plot $y=e^{-0.05 x}[5 \sin 4(x-0.39)]+10$ on a graphing calculator with the original model for $0 \leq x \leq 10$. State how this model differs from the original model, and how it might account for Cassandra's observation.

For the trigonometric function, determine the vertical stretch or compression and the horizontal stretch or compression. Then, plot the function and ifnd its period.

$$y=\frac{1}{3} \sin 2 x$$

Determine the new function after the given transformation.

$$q(x)=-0.5(x+2)^2-12$$

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Do transformed functions of the form $f(x)=a \tan \left(\frac{1}{b} x\right)$ have the same behavior seen in the parent function, that is, f(-x)=-f(x) ? Justify your answer and how it helps to graph transformed functions.

Graph one cycle of the transformed tangent function g(x) together with the parent function f(x)=tan x. g(x)=-$\frac{1}{4} \tan 4 x$

Determine the new function after the given transformation.

$$p(x)=0.1(x-8)^2+9$$

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Determine the period and the midline of the graph, and where the graph crosses the midline. For a sine or cosine function, find the amplitude and the maximum and minimum values and where they occur. For a tangent function, find the asymptotes and the values of the halfway points. Then plot one cycle. f(x)=3 sin $\frac{\pi}{2}(x-2)$+3

Graph one cycle of the transformed tangent function g(x) together with the parent function f(x)=tan x. g(x)=-$\tan \frac{x}{3}$

Explain why is the vertical stretch parameter, a, referred to as the amplitude when transforming sine and cosine functions, but not tangent functions. If it is not the amplitude, explain how it can be used to help graph the function.

In photosynthesis, a plant converts carbon dioxide and water to sugar and oxygen. This process is studied by measuring a plant's carbon assimilation C (in micromoles of CO$_2$ per square meter per second). For a bean plant, C(t)=1.2 $\sin \frac{\pi}{12}(t-6)+7$, where t is time in hours starting at midnight.

a. Determine the period of the function?

b. Find the maximum and the time it occurs.

State the key features of the graphs of a trigonometric function, and how you can use a trigonometric function to model real-world data.

State a rotation that will map the graph of $f(\theta)=\sin \theta$ onto itself on the interval $0 \leq \theta \leq 2 \pi$.

For the situation described: (a) Plot the data, roughly fit a sine model of the form $y=a \sin \frac{1}{b}(x-h)+k$ to the data, and plot the model on the same grid as the data.

(b) Determine a sine regression model using a graphing calculator, and then visually compare the two models in relation to the data.

$(c)$ Use a graphing calculator and the regression model to answer the question.

The table shows the decimal portion of the moon's surface illuminated from Earth's view at midnight from Washington, D.C., in March, 2014, where x represents the day of the month.

A "gibbous moon" refers to the moon phases when more than half of the moon is illuminated, but it is not a full moon. Determine about how long is the period that the moon is either a gibbous moon or a full moon.

For the trigonometric function, determine the vertical stretch or compression and the horizontal stretch or compression. Then, plot the function and ifnd its period.

$$y=-2 \cos \frac{1}{3} x$$